Theorem dvdsrd 14339

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
dvdsrvald.1	|- ( ph -> B = ( Base ` R ) )
dvdsrvald.2	|- ( ph -> .|| = ( ||r ` R ) )
dvdsrvald.r	|- ( ph -> R e. SRing )
dvdsrvald.3	|- ( ph -> .x. = ( .r ` R ) )

Assertion

Ref	Expression
dvdsrd	|- ( ph -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )

Distinct variable groups:   z, B   z, X   z, Y   z, R   z, .x.   ph, z

Allowed substitution hint:   .|| ( z)

Proof of Theorem dvdsrd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsrvald.r	. . . . . 6 |- ( ph -> R e. SRing )
2		reldvdsrsrg 14337	. . . . . 6 |- ( R e. SRing -> Rel ( ||r ` R ) )
3	1, 2	syl 14	. . . . 5 |- ( ph -> Rel ( ||r ` R ) )
4		dvdsrvald.2	. . . . . 6 |- ( ph -> .|| = ( ||r ` R ) )
5	4	releqd 4839	. . . . 5 |- ( ph -> ( Rel .|| <-> Rel ( ||r ` R ) ) )
6	3, 5	mpbird 167	. . . 4 |- ( ph -> Rel .|| )
7		brrelex12 4793	. . . 4 |- ( ( Rel .|| /\ X .|| Y ) -> ( X e. _V /\ Y e. _V ) )
8	6, 7	sylan 283	. . 3 |- ( ( ph /\ X .|| Y ) -> ( X e. _V /\ Y e. _V ) )
9	8	ex 115	. 2 |- ( ph -> ( X .|| Y -> ( X e. _V /\ Y e. _V ) ) )
10		simplr 529	. . . . . 6 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> X e. B )
11	10	elexd 2829	. . . . 5 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> X e. _V )
12		simprr 533	. . . . . . 7 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> ( z .x. X ) = Y )
13	1	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> R e. SRing )
14		simprl 531	. . . . . . . . . 10 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> z e. B )
15		dvdsrvald.1	. . . . . . . . . . 11 |- ( ph -> B = ( Base ` R ) )
16	15	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> B = ( Base ` R ) )
17	14, 16	eleqtrd 2313	. . . . . . . . 9 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> z e. ( Base ` R ) )
18	10, 16	eleqtrd 2313	. . . . . . . . 9 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> X e. ( Base ` R ) )
19		eqid 2234	. . . . . . . . . 10 |- ( Base ` R ) = ( Base ` R )
20		eqid 2234	. . . . . . . . . 10 |- ( .r ` R ) = ( .r ` R )
21	19, 20	srgcl 14213	. . . . . . . . 9 |- ( ( R e. SRing /\ z e. ( Base ` R ) /\ X e. ( Base ` R ) ) -> ( z ( .r ` R ) X ) e. ( Base ` R ) )
22	13, 17, 18, 21	syl3anc 1274	. . . . . . . 8 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> ( z ( .r ` R ) X ) e. ( Base ` R ) )
23		dvdsrvald.3	. . . . . . . . . 10 |- ( ph -> .x. = ( .r ` R ) )
24	23	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> .x. = ( .r ` R ) )
25	24	oveqd 6075	. . . . . . . 8 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> ( z .x. X ) = ( z ( .r ` R ) X ) )
26	22, 25, 16	3eltr4d 2318	. . . . . . 7 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> ( z .x. X ) e. B )
27	12, 26	eqeltrrd 2312	. . . . . 6 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> Y e. B )
28	27	elexd 2829	. . . . 5 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> Y e. _V )
29	11, 28	jca 306	. . . 4 |- ( ( ( ph /\ X e. B ) /\ ( z e. B /\ ( z .x. X ) = Y ) ) -> ( X e. _V /\ Y e. _V ) )
30	29	rexlimdvaa 2663	. . 3 |- ( ( ph /\ X e. B ) -> ( E. z e. B ( z .x. X ) = Y -> ( X e. _V /\ Y e. _V ) ) )
31	30	expimpd 363	. 2 |- ( ph -> ( ( X e. B /\ E. z e. B ( z .x. X ) = Y ) -> ( X e. _V /\ Y e. _V ) ) )
32	15, 4, 1, 23	dvdsrvald 14338	. . . . . 6 |- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
33	32	adantr 276	. . . . 5 |- ( ( ph /\ ( X e. _V /\ Y e. _V ) ) -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
34	33	breqd 4125	. . . 4 |- ( ( ph /\ ( X e. _V /\ Y e. _V ) ) -> ( X .|| Y <-> X { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } Y ) )
35		simpl 109	. . . . . . . 8 |- ( ( x = X /\ y = Y ) -> x = X )
36	35	eleq1d 2303	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( x e. B <-> X e. B ) )
37	35	oveq2d 6074	. . . . . . . . 9 |- ( ( x = X /\ y = Y ) -> ( z .x. x ) = ( z .x. X ) )
38		simpr 110	. . . . . . . . 9 |- ( ( x = X /\ y = Y ) -> y = Y )
39	37, 38	eqeq12d 2249	. . . . . . . 8 |- ( ( x = X /\ y = Y ) -> ( ( z .x. x ) = y <-> ( z .x. X ) = Y ) )
40	39	rexbidv 2545	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( E. z e. B ( z .x. x ) = y <-> E. z e. B ( z .x. X ) = Y ) )
41	36, 40	anbi12d 473	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( ( x e. B /\ E. z e. B ( z .x. x ) = y ) <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
42		eqid 2234	. . . . . 6 |- { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) }
43	41, 42	brabga 4387	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> ( X { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
44	43	adantl 277	. . . 4 |- ( ( ph /\ ( X e. _V /\ Y e. _V ) ) -> ( X { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
45	34, 44	bitrd 188	. . 3 |- ( ( ph /\ ( X e. _V /\ Y e. _V ) ) -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
46	45	ex 115	. 2 |- ( ph -> ( ( X e. _V /\ Y e. _V ) -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) ) )
47	9, 31, 46	pm5.21ndd 713	1 |- ( ph -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   E.wrex 2523   _Vcvv 2815   class class class wbr 4114   {copab 4175   Rel wrel 4759   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375  SRingcsrg 14206   ||_rcdsr 14330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mgp 14160  df-srg 14207  df-dvdsr 14333

This theorem is referenced by:  dvdsr2d  14340  dvdsrmuld  14341  dvdsrcld  14342  dvdsrcl2  14344  dvdsrtr  14346  dvdsrmul1  14347  opprunitd  14355  crngunit  14356  rhmdvdsr  14420  subrgdvds  14481  cnfldui  14863